Quantized Algebra and Physics

The following sections are included:

• PREFACE

• CONTENTS

• Programs

The following sections are included:

• Introduction

• Brauer-Schur functions

• Cauchy type formula

• Monomial expansion

• References

This paper studies -operators and the role they play in relating diverse objects such as Rota-Baxter operators, associative Yang-Baxter equations and dendriform algebras. The paper gives a survey of recent developments. It also presents new results on antisymmetric infinitesimal bialgebras, discusses the relationship of -operators with Rota-Baxter operators and relative differential operators, provides further characterizations of dendriform algebras and constructs solutions of associative Yang-Baxter equations from dendriform algebras.

In this paper, we used the free fields to construct a class of irreducible modules of the affine Lie algebra .

In this paper, we first study properties of generic exp-polynomial functions on ℤⁿ. Then we use these properties to deduce a criterion for Verma modules to be irreducible over generic exp-polynomial Lie algebras relative to a total order.

In this paper it is shown under mild assumptions that the local solvability of an infinite dimensional formal Cauchy problem is equivalent to a set of zero curvature relations. The role this type of Cauchy problems plays in integrable systems is illustrated at the hand of lower triangular Toda hierarchies.

Let l, n ∈ ℕ and . Let V be the natural representation of the quantized enveloping algebra and the specialized Birman–Murakami–Wenzl algebra with parameters depending on (see (2) and Section 2 for their definitions). There is a Schur–Weyl duality between and on V^⊗n. We prove that V^⊗n can be decomposed into a direct sum of the subspaces of q-partially harmonic tensors of different valences and there is also a Schur–Weyl duality between the Hecke algebra of type A_n-1 and on the subspaces of q-harmonic tensors. We construct explicitly a complete set of maximal vectors in V^⊗n and identify some simple -modules that they generate with the simple -modules arising from Enyang's cellular basis of .

We give an exact solution to the nonlinear optimization problem of approximating a Hermitian matrix by positive semi-definite matrices. Our algorithm was then used to judge whether a quantum state is entangled or not. We show that the exact approximation of a density matrices by tensor product of positive semi-definite operators is determined by the additivity property of the density matrix.

The 2-partitions of the set of positive roots and of the root system of finite type or affine type are completely classified.

The theory of weak Hopf algebras is a development of the theory of Hopf algebras, which is important in the theory of quantum groups and the related theory of mathematical physics. In this paper, we give a survey of this theory and its applications.

Firstly, we review some concepts related to weak Hopf algebras and their relationship and moreover, some algebraical structures. Secondly, we collect the generalizations of quantum double, i.e. quantum quasi-double and quantum G-double, constructed from some certain weak Hopf algebras so as to obtain a class of singular solutions of the quantum Yang-Baxter equation (QYBE). Their properties are also introduced in this part. Thirdly, as the important examples of weak Hopf algebras, some weaken quantized enveloping algebra are given which are associated with some generalizations of Lie algebra, such as generalized Kac-Moody algebra, Borcherds superalgebras. Moreover, their algebraic structures are discussed such that one can understand them clearly. Finally, some generalized notions of tensor category are introduced including pre-tensor category and weak tensor category and their braidings. They can be realized respectively from the categories of representations of the related classes of weak Hopf algebras and their quantum quasi-doubles. As the categorical version of a (singular) solutions of the QYBE, a general Yang-Baxter operator is studied. In summary, we list the diagrams of relations among these kinds of categories.

We show that the quantum algebra U_q(f(k)) has a presentation with generators x^±1, y, z and relations